Topological factoring onto odometers and speedups of Toeplitz flows

A recent paper of Ash, Dykstra, and LeMasurier asks if bounded speedups of Toeplitz flows must be Toeplitz flows. In this paper, we answer this question in the negative by finding, for any Toeplitz flow generated by a primitive, proper, constant-length substitution rule, a speedup which is not Toeplitz. This investigation leads to a characterization of the Cantor minimal systems that factor onto an odometer and those which are topologically conjugate to an odometer, yielding topological versions of results of Foreman, Gao, Hill, Silva, and Weiss in the measure-preserving category.